OCR MEI Further Pure Core 2019 June — Question 14 13 marks

Exam BoardOCR MEI
ModuleFurther Pure Core (Further Pure Core)
Year2019
SessionJune
Marks13
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
Topic3x3 Matrices
TypeParameter values for unique solution
DifficultyChallenging +1.2 This is a standard Further Maths question on systems of linear equations and geometric interpretation of planes. Part (a) requires finding when the determinant is zero (routine calculation) and understanding sheaf geometry. Part (b) involves solving a 3×3 system with a parameter, which is methodical but requires careful algebra. While it's a multi-part question requiring several techniques, the concepts are core Further Maths material with no novel insights needed—moderately above average difficulty.
Spec4.03s Consistent/inconsistent: systems of equations
14 Three planes have equations $$\begin{aligned} - x + a y & = 2 \\ 2 x + 3 y + z & = - 3 \\ x + b y + z & = c \end{aligned}$$ where \(a\), \(b\) and \(c\) are constants.
  1. In the case where the planes do not intersect at a unique point,
    1. find \(b\) in terms of \(a\),
    2. find the value of \(c\) for which the planes form a sheaf.
  2. In the case where \(b = a\) and \(c = 1\), find the coordinates of the point of intersection of the planes in terms of \(a\).
Question 14:
AnswerMarks Guidance
14(a) (i)
 
let M = 2 3 1
 
 
AnswerMarks Guidance
 1 b 1 M1 3.1a
coefficients
AnswerMarks Guidance
det M = b  3  aB1 1.1b
det M = 0
AnswerMarks Guidance
 b = a + 3det M = 0 M1
3.2a1.1b
 b = a + 3A1 3.2a
[4]
AnswerMarks Guidance
14(a) (ii)
(a + b)y + z = c + 2, b = a+3
 (2a+3)y + z = c+2
AnswerMarks
 c = 1M1
M1
A1cao
AnswerMarks
[3]3.1a
3.1a
AnswerMarks
3.2areduce system to 2 equations
in 2 variables
use b = a + 3 to find value of
AnswerMarks
c for consistencyone including c
or other valid method
AnswerMarks Guidance
14(b) 3a a a 
1 
M1 =  1 1 1
 
3
 
AnswerMarks
2a3 2a 32aM1
A2
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M13.1a
1.1b
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1.1battempt to find M1
A1 any 6 entries correct
 1/their det
 2   62a 
  1 
M1 3  2
   
  3 
AnswerMarks Guidance
1  4a9M1 1.1b
62a 2 4a9
coordinates are ( , , )
AnswerMarks Guidance
3 3 3A1cao 3.2a
[6]
M1
1.1b
pre-multiplying by their M1
Alternative solution
x + ay = 2, x + ay + z = 1
 2x + z = 1, z = 2x 1
x + ay = 2  y = (2+ x)/a
3x6
 2x 12x3
a
2a6 2 4a9
x y , z = 
AnswerMarks
3 3 3M1
M1
M1
A3
AnswerMarks
[6]from 2 equations
to get eqn in one unknown
eliminate one variable
eliminate another variable
substitute into 3rd eqn
Question 14:
14 | (a) | (i) | 1 a 0
 
let M = 2 3 1
 
 
 1 b 1  | M1 | 3.1a | finding matrix of
coefficients
det M = b  3  a | B1 | 1.1b
det M = 0
 b = a + 3 | det M = 0 | M1 | 1.1b
3.2a | 1.1b
 b = a + 3 | A1 | 3.2a
[4]
14 | (a) | (ii) | x = ay  2  (2a+3)y + z = 1
(a + b)y + z = c + 2, b = a+3
 (2a+3)y + z = c+2
 c = 1 | M1
M1
A1cao
[3] | 3.1a
3.1a
3.2a | reduce system to 2 equations
in 2 variables
use b = a + 3 to find value of
c for consistency | one including c
or other valid method
14 | (b) | 3a a a 
1 
M1 =  1 1 1
 
3
 
2a3 2a 32a | M1
A2
M1 | 3.1a
1.1b
1.1b | attempt to find M1
A1 any 6 entries correct
 1/their det
 2   62a 
  1 
M1 3  2
   
  3 
1  4a9 | M1 | 1.1b | pre-multiplying by their M1
62a 2 4a9
coordinates are ( , , )
3 3 3 | A1cao | 3.2a | accept in vector form
[6]
M1
1.1b
pre-multiplying by their M1
Alternative solution
x + ay = 2, x + ay + z = 1
 2x + z = 1, z = 2x 1
x + ay = 2  y = (2+ x)/a
3x6
 2x 12x3
a
2a6 2 4a9
x y , z = 
3 3 3 | M1
M1
M1
A3
[6] | from 2 equations
to get eqn in one unknown
eliminate one variable
eliminate another variable
substitute into 3rd eqn
14 Three planes have equations

$$\begin{aligned}
- x + a y & = 2 \\
2 x + 3 y + z & = - 3 \\
x + b y + z & = c
\end{aligned}$$

where $a$, $b$ and $c$ are constants.
\begin{enumerate}[label=(\alph*)]
\item In the case where the planes do not intersect at a unique point,
\begin{enumerate}[label=(\roman*)]
\item find $b$ in terms of $a$,
\item find the value of $c$ for which the planes form a sheaf.
\end{enumerate}\item In the case where $b = a$ and $c = 1$, find the coordinates of the point of intersection of the planes in terms of $a$.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Pure Core 2019 Q14 [13]}}
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